Edge-graceful labeling
In graph theory, an edge-graceful graph labeling is a type of graph labeling. This is a labeling for simple graphs in which no two distinct edges connect the same two distinct vertices, no edge connects a vertex to itself, and the graph is connected. Edge-graceful labelings were first introduced by S. Lo in his seminal paper.
Given a graph G, we denote the set of edges by E(G) and the vertices by V(G). Let q be the cardinality of E(G) and p be that of V(G). Once a labeling of the edges is given, a vertex u of the graph is labeled by the sum of the labels of the edges incident to it, modulo p. Or, in symbols, the induced labeling on the vertex u is given by
where V(u) is the label for the vertex and E(e) is the assigned value of an edge incident to u.
The problem is to find a labeling for the edges such that all the labels from 1 to q are used once and the induced labels on the vertices run from 0 to p − 1. In other words, the resulting set for labels of the edges should be and for the vertices.
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